Rayleigh-Ritz procedure

Two modifications of the Rayleigh-Ritz procedure were found useful in the present work. Expanding the gradients in Equation 14.3 to first-order in a Taylor series about X gives... 

It has been shown that many differential equations that originate from the physical sciences have equivalent variational formulations." This is the basis for the well-known Rayleigh-Ritz procedure which in turn forms the basis for the finite element methods. 

Dunham [5] derived these expressions Y ((t)e,Be,a0, necessarily manually, through a JBKW procedure, which he claimed to make more general [4] than what had appeared in previous literature. Dunham reported expressions F containing coefficients aj up to a, and Sandeman [19] and Woolley [20] extended manually these results according to a roughly analogous procedure. Kilpatrick [21] applied perturbation theory in successive orders to derive expressions for 1, and Bouanich [22] applied Rayleigh-Ritz perturbation theory for solution of... 

Instead of applying tail cancellation as in Sect.2.1 where we derived the KKR-ASA equations, one may use the linear combination of muffin-tin orbitals (5.27) directly in a variational procedure. This has the advantages that it leads to an eigenvalue problem and that it is possible to include non-muffin-tin perturbations to the potential. According to the Rayleigh-Ritz variational principle, one varies y to make the energy functional stationary, i.e. 

Wheeler and collaborators [3], in the context of nuclear physics, showed at that time that the limit in the variational procedure potential itself was not reached. Indeed, the Rayleigh-Ritz (RR) variational scheme teaches us how to obtain the best value for a parameter in a trial function, i.e., exponents of Slater (STO) or Gaussian (GTO) type orbital, Roothaan or linear combination of atomic orbitals (LCAO) expansion coefficients and Cl coefficients. Instead, the generator coordinate method (GCM) introduces the Hill-Wheeler (HW) equation, an integral transform algorithm capable, in principle, to find the best functional form for a given trial function. We present the GCM and the HW equation in Section 2. 

This procedure is referred to as the Rayleigh-Ritz method, which is equivalent to the Galerkin method. 

An alternative approach, utilizing the same wavefunction representation, was used by Handy (1996) in implementing a Rayleigh-Ritz variational procedure with respect to the fi dependence of the a -coefficients. 

An important observation about the EMM approach is that it is manifestly a scale-translation (affine map) invariant variational procedure, unlike other approaches, such as Rayleigh-Ritz. At each order, the variation with respect to the Cj s is actually optimizing over all possible affine map transforms of the polynomial sampling function. 

---